Model
A model is something defined by a model. So the concept or type of a model is impredicative or "circular"; given certain assumptions, the impredicativity of the type allows us to use models to define and characterize a theory. In fact models are the principal mechanisms by which we evaluate or interpret theories, as they are the means by which we associate properties and structures to a theory. The prototypical example of this is "truth" in a logical model, where we say that a given structure models a theory A \models \mathbb{T} when the sentences of the theory are "true" in the structure. Note that a model is "something defined", and that this is an explicit part of its definition. So it is a particular example of a presentation (the result of an operator) or sentence (the result of a hypothesis). In fact, models are "the most particular" examples of presentations and sentences. Why? To illustrate the role of impredicativity, let us construct a particular formalism where we assume a strict connection between operators and presentations and that, more specifically, any operator that generates or defines a model is also impredicative, i.e. invokes, acts upon, or "follows" itself. Further, let us reject all impredicative operators and hypotheses. Then, in that formalism, we can say that models are explicitly undefined. This sequence of assumptions is quite important if we want to characterize a theory "as it really is", because it allows us to abstract out the particular—and thus arbitrary—features of models when we use them to study a theory. Under the provisional formalism, the concept of a model is written as R^\infty . Truth and predicativity See the article on logical models. Models describe properties of sentences. Often, we use them to describe the "truth" property of a sentence via a predicative function, which maps the sentence to a set of truth values. Thus, one may ask, how does one determine the truth of a model? Following Tarski: only by appeal to another model. So the undefinability of truth is the simplest, most "classical" reason for understanding the concept of a model as impredicative. By contrast, one can reject impredicativity in order to construct a ramified theory... Idempotence A idempotent correspondence \mathcal{G} is precisely one that defines or constructs itself, written \mathcal{G} : X \to \mathcal{G} where X is understood to be defined. As a homotopy type The concept of a model, written R^\infty , is a homotopy type of homotopy level \infty or, for short, a \infty -type. This means that it is an \infty -groupoid and thus may be viewed as a topological space up to weak homotopy equivalence. As a class of operators Recall that we said a model is explicitly undefined. Under the domain formalism, this means that the concept of a model is the class of operators (given some theory) from an defined domain to an undefined codomain, i.e. the class of all implications given some theory. As a manifold Recall that as part of the definition, a model is something defined. Assume that we can cleanly parse these different parts of the definition. Then given a topological assumption on hypotheses, we can interpret the assumption that "a model is something defined" as a local assumption on points (with the type or concept of the model standing in as the global space) and then, via the parthood relation in the definition, construct models as manifolds.